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Five limit

The 5-limit parent comma for the schismatic family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its monzo is |-15 8 1>, and flipping that yields <<1 -8 -15|| for the wedgie. This tells us the generator is a fifth and that we will need eight fourths in succession to reach the pitch class of a major third. In fact, 10 = (4/3)^8 * 32805/32768.

The 5-limit version of the temperament is a microtemperament, sometimes called Helmholtz or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. 53edo is a possible tuning for schismatic, but you need 118edo if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering.

POTE generator: ~3/2 = 701.736
Mapping generator: ~3

Map: [<1 0 15|, <0 1 -8|]
EDOs: 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bc

Seven limit children

The second comma of the normal comma list defines which 7-limit family member we are looking at. Adding |25 -14 0 -1> gives garibaldi, |-44 26 0 1> grackle, |6 -2 0 -1> schism and |-59 39 0 -1> pontiac; these all have a fifth as generator. Bischismic adds |-69 40 0 2> and has a fifth generator with a half-octave period. Guiron adds 1029/1024 = |-10 1 0 3>, with an 8/7 generator, three of which give the fifth, and term adds |-94 54 0 3> with a 1/3 octave period. Sesquiquartififths adds |-35 15 0 4> and slices the fifth in four.

Garibaldi

Commas: 225/224, 3125/3087

7-limit minimax tuning:
7-limit: [|1 0 0 0>, |5/3 1/15 0 -1/15>,
|5/3 -8/15 0 8/15>, |5/3 -14/15 0 14/15>]
Eigenmonzos: 2, 7/6

9-limit: [|1 0 0 0>, |25/16 1/8 0 -1/16>,
|5/2 -1 0 1/2>, |25/8 -7/4 0 7/8>]
Eigenmonzos: 2, 9/7

POTE generator: ~3/2 = 702.085
Mapping generator: ~3

Map: [<1 0 15 25|, <0 1 -8 -14|]
Wedgie: <<1 -8 -14 -15 -25 -10||
EDOs: 12, 29, 41, 53, 94, 241c, 335cd, 576cd
Badness: 0.0216

11-limit

Commas: 225/224, 385/384, 2200/2187

Minimax tuning:
[|1 0 0 0 0>, |25/16 1/8 0 -1/16 0>, |5/2 -1 0 1/2 0>,
|25/8 -7/4 0 7/8 0>, |47/16 23/8 0 -23/16 0>]
Eigenmonzos: 2, 9/7

POTE generator: ~3/2 = 702.157
Mapping generator: ~3

Map: [<1 0 15 25 -33|, <0 1 -8 -14 23|]
Edos: 41, 53, 94, 135, 229c, 323c, 417ce
Badness: 0.0274

13-limit

Commas: 225/224, 275/273, 325/324, 385/384

POTE generator: ~3/2 = 702.113
Mapping generator: ~3

Map: [<1 0 15 25 -33 -28|, <0 1 -8 -14 23 20|]
EDOs: 12, 41, 53, 94, 429cdef, 523cdef
Badness: 0.0207

Cassandra

Commas: 100/99, 225/224, 245/242

POTE generator: ~3/2 = 702.321
Mapping generator: ~3

Map: [<1 0 15 25 32|, <0 1 -8 -14 -18|]
EDOs: 12, 29, 41, 123c, 217ce, 258ce
Badness: 0.0236

13-limit Cassandra

Commas: 100/99, 105/104, 196/195, 245/242

POTE generator: ~3/2 = 702.559
Mapping generator: ~3

Map: [<1 0 15 25 32 37|, <0 1 -8 -14 -18 -21|]
EDOs: 12, 29, 41, 152cdf, 193cdf, 234cdf
Badness: 0.0207

Helenus

Commas: 99/98 176/175 3125/3087

POTE generator: ~3/2 = 701.725
Mapping generator: ~3

Map: [<1 0 15 25 51|, <0 1 -8 -14 -30|]
EDOs: 12, 53, 118d, 171de
Badness: 0.0356

13-limit Helenus

Commas: 99/98, 176/175, 275/273, 847/845

POTE generator: ~3/2 = 701.747
Mapping generator: ~3

Map: [<1 0 15 25 51 56|, <0 1 -8 -14 -30 -33|]
EDOs: 53, 118d, 171de
Badness: 0.0263

Hemigari

Commas: 121/120, 225/224, 3125/3087

POTE generator: ~63/55 = 248.918
Mapping generator: ~110/63

Map: [<1 0 15 25 9|, <0 2 -16 -28 -7|]
EDOs: 29, 53, 82e, 135e, 188ce
Badness: 0.0507

13-limit

Commas: 121/120, 169/168, 225/224, 275/273

POTE generator: ~15/13 = 248.918
Mapping generator: ~26/15

Map: [<1 0 15 25 9 14|, <0 2 -16 -28 -7 -13|]
EDOs: 29, 53, 82e, 135ef, 188cef
Badness: 27.464

Guiron

Commas: 1029/1024, 10976/10935

Minimax tuning:
7+9 limit: [|1 0 0 0>, |15/8 0 -1/8 0>,
|0 0 1 0>, |65/24 0 1/24 0>]
Eigenmonzos: 2, 5/4

POTE generator: ~8/7 = 233.930
Mapping generator: ~8/7

Map: [<1 1 7 3|, <0 3 -24 -1|]
Wedgie: <<3 -24 -1 -45 -10 65||
EDOs: 36, 41, 77, 118, 277d
Badness: 0.0475

11-limit

Commas: 385/384, 441/440, 10976/10935

Minimax tuning:
[|1 0 0 0 0>, |15/8 0 -1/8 0 0>,
|0 0 1 0 0>, |65/24 0 1/24 0 0>,
|37/6 0 -7/6 0 0>]
Eigenmonzos: 2, 5/4

POTE generator: ~8/7 = 233.931
Mapping generator: ~8/7

Map: [<1 1 7 3 -2|, <0 3 -24 -1 28|]
Edos: 41, 77, 118, 159, 200, 277d
Badness: 0.0266

13-limit

Commas: 196/195, 352/351, 385/384, 729/728

POTE generator: ~8/7 = 233.890
Mapping generator: ~8/7

Map: [<1 1 7 3 -2 0|, <0 3 -24 -1 28 19|]
EDOs: 41, 77, 118
Badness: 0.0284

Pogo

Commas: 32805/32768, 118098/117649

POTE generator: ~9/7 = 433.901
Mapping generator: ~9/7

Map: [<2 1 22 2|, <0 3 -24 5|]
EDOs: 36, 94, 130, 171, 183, 224, 354
Badness: 0.0796

11-limit

Commas: 540/539, 4000/3993, 32805/32768

POTE generator: ~9/7 = 433.911
Mapping generator: ~9/7

Map: [<2 1 22 2 25|, <0 3 -24 5 -25|]
EDOs: 36, 94, 130, 224, 354, 578
Badness: 0.0319

13-limit

Commas: 540/539, 729/728, 4000/3993, 4225/4224

POTE generator: ~9/7 = 433.911

Map: [<2 1 22 2 25 -2|, <0 3 -24 5 -25 13|]
EDOs: 36, 94, 130, 224, 354, 578
Badness: 0.0175

Sanjaab

Commas: 225/224, 3125/3087, 1331/1323

POTE generator: ~11/10 = 165.974

Map: [<1 2 -1 -3 0|, <0 -3 24 42 25|]
EDOs: 29, 65d, 94, 441cde, 535cde, 629cde
Badness: 0.0580

13-limit

Commas: 225/224, 275/273, 847/845, 1331/1323

POTE generator: ~11/10 = 165.963

Map: [<1 2 -1 -3 0 -1|, <0 -3 24 42 25 34|]
EDOs: 29, 65d, 94
Badness: 0.0338

Squirrel

Commmas: 686/675, 32805/32768

POTE generator: ~160/147 = 166.140

Map: [<1 2 -1 1|, <0 -3 24 13|]
EDOs: 29, 36, 65
Badness: 0.1747

11-limit

Commas: 245/242, 686/675, 896/891

POTE generator: ~11/10 = 166.097

Map: [<1 2 -1 1 0|, <0 -3 24 13 25|]
EDOs: 29, 36, 65
Badness: 0.0683

13-limit

Commas: 91/90, 169/168, 245/242, 896/891

POTE generator: ~11/10 = 166.054

Map: [<1 2 -1 1 0 3|, <0 -3 24 13 25 5|]
EDOs: 29, 36, 65f, 94df, 159df
Badness: 0.0437

Schism

Commas: 64/63, 360/343

POTE generator: ~3/2 = 701.556
Mapping generator: ~3

Map: [<1 0 15 6|, <0 1 -8 -2|]
Wedgie: <<1 -8 -2 -15 -6 18||
EDOs: 12, 41d, 53d
Badness: 0.0566

11-limit

Commas: 45/44, 64/63, 99/98

POTE generator ~3/2 = 702.136
Mapping generator: ~3

Map: [<1 0 15 6 13|, <0 1 -8 -2 -6|]
EDOs: 12, 29de, 41de
Badness: 0.0375

Pontiac

Commas: 32805/32768, 4375/4374

7-limit minimax:
[|1 0 0 0>, |74/47 0 -1/47 1/47>, |113/47 0 8/47 -8/47>,
|113/47 0 -39/47 39/47>]
Eigenmonzos: 2, 7/5

9-limit minimax:
[|1 0 0 0>, |3/2 1/5 -1/10 0>,
|3 -8/5 4/5 0>, |-1/2 39/5 -39/10 0>]
Eigenmonzos: 2, 10/9

POTE generator: 701.757
Mapping generator: ~3

Map: [<1 0 15 -59|, <0 1 -8 39|]
Wedgie: <<1 -8 39 -15 59 113||
Edos: 53, 118, 171, 1079, 1250, 1421
Badness: 0.0141

Grackle

Commas: 126/125, 32805/32768

7-limit minimax
Eigenmonzos: 2, 7/6

9-limit minimax
Eigenmonzos: 2, 9/7

POTE generator: 701.239
Mapping generator: ~3

Map: [<1 0 15 -44|, <0 1 -8 -26|]
Wedgie: <<1 -8 -26 -15 -44 -38||
Edos: 77, 89, 101, 166c, 243c
Badness: 0.070

Bischismic

Commas: 3136/3125, 32805/32768

7-limit minimax
Eigenmonzos: 2, 7/6

9-limit minimax
Eigenmonzos: 2, 9/7

POTE generator: 701.592
Mapping generator: ~3

Map: [<2 0 30 69|, <0 1 -8 -20|]
Wedgie: <<2 -16 -40 -30 -69 -48||
Edos: 118, 130, 248, 378, 508
Badness: 0.0547

11-limit

Commas: 441/440, 3136/3125, 8019/8000

POTE generator: ~3/2 = 701.612
Mapping generator: ~3

Map: [<2 0 30 69 102|, <0 1 -8 -20 -30|]
EDOs: 12, 118, 130, 248
Badness: 0.0282

13-limit

Commas: 441/440, 729/728, 1001/1000, 3136/3125

POTE generator: ~3/2 = 701.590
Mapping generator: ~3

Map: [<2 0 30 69 102 -75|, <0 1 -8 -20 -30 26|]
EDOs: 12, 118, 130, 248, 378
Badness: 0.0287

Kleischismic

Commas: 32805/32768, 1500625/1492992

POTE generator: ~36/35 = 50.920
Mapping generator: ~35/24

Map: [<2 1 22 -15|, <0 2 -16 19|]
Wedgie: <<4 -32 38 -60 49 178||
EDOs: 24, 94, 118, 212, 330, 542d, 872cd
Badness: 0.1106

11-limit

Commas: 385/384, 9801/9800, 14641/14580

POTE generator: ~36/35 = 50.918
Mapping generator: ~16/11

Map: [<2 1 22 -15 8|, <0 2 -16 19 -1|]
EDOs: 24, 94, 118, 212, 330e, 542de
Badness: 0.0367

13-limit

Commas: 352/351, 385/384, 729/728, 1575/1573

POTE generator: ~36/35 = 50.938
Mapping generator: ~16/11

Map: [<2 1 22 -15 8 15|, <0 2 -16 19 -1 -7|]
EDOs: 24, 94, 118, 212f
Badness: 0.0376

Term

Commas: 32805/32768, 250047/250000

7-limit minimax
Eigenmonzos: 2, 6/5

9-limit minimax
Eigenmonzos: 2, 9/7

POTE generator: ~3/2 = 701.742
Mapping generator: ~3

Map: [<3 0 45 94|, <0 1 -8 -18|]
Wedgie: <<3 -24 -54 -45 -94 -58||
Edos: 171, 1038, 1209
Badness: 0.0200

Sesquiquartififths

Commas: 2401/2400, 32805/32768

7-limit minimax
Eigenmonzos: 2, 7/6

9-limit minimax
Eigenmonzos: 2, 9/7

POTE generator: ~448/405 = 175.434
Mapping generator: ~448/405

Map: [<1 1 7 5|, <0 4 -32 -15|]
Wedgie: <<4 -32 -15 -60 -35 55||
Edos: 171, 643, 2100edo
Badness: 0.0112

Sesquart

Commas: 243/242, 441/440, 16384/16335

POTE generator: ~256/231 = 175.406
Mapping generator: ~256/231

Map: [<1 1 7 5 2|, <0 4 -32 -15 10|]
EDOs: 41, 89, 130
Badness: 0.0293

13-limit

Commas: 243/242, 364/363, 441/440, 105644/105625

POTE generator: ~256/231 = 175.409

Map: [<1 1 7 5 2 -2|, <0 4 -32 -15 10 39|]
EDOs: 41, 89, 130, 301e, 431e
Badness: 0.0224

Bisesqui

Commas: 2401/2400, 9801/9800, 32805/32768

POTE generator: ~448/405 = 175.435

Map: [<2 2 14 10 23|, <0 4 -32 -15 -55|]
EDOs: 130, 212, 342, 1156, 1498
Badness: 0.0170

Sextilififths

Commas: 32768/32805, 235298/234375

POTE generator: ~21/20 = 83.053
Mapping generator: ~21/20

Map: [<1 2 -1 -1|, <0 -6 48 55|]
EDOs: 29, 101, 130, 289, 419
Badness: 0.1088

11-limit

Commas: 441/440, 4000/3993, 235298/234375

POTE generator: ~21/20 = 83.049
Mapping generator: ~21/20

Map: [<1 2 -1 -1 0|, <0 -6 48 55 50|]
EDOs: 29, 130, 289
Badness: 0.0455

13-limit

Commas: 363/363, 441/440, 676/675, 10985/10976

POTE generator: ~21/20 = 83.049
Mapping generator: ~21/20

Map: [<1 2 -1 -1 0 1|, <0 -6 48 55 50 39|]
EDOs: 29, 130, 289
Badness: 0.0253

Tsaharuk

Commas: 32805/32768, 420175/419904

POTE generator: ~243/224 = 140.350
Mapping generator: ~243/224

Map: [<1 1 7 0|, <0 5 -40 24|]
Wedgie: <<5 -40 24 -75 24 168||
EDOs: 17, 26, 43, 60, 77, 94, 171
Badness: 0.0307

Quanharuk

Commas: 16875/16807, 32805/32768

POTE generator: ~56/45 = 380.355
Mapping generator: ~56/45

Map: [<1 0 15 12|, <0 5 -40 -29|]
Wedgie: <<5 -40 -29 -75 -60 45||
EDOs: 41, 142, 183, 224, 1303d, 1527cd, 1751cd, 1975cd
Badness: 0.0720

11-limit

Commas: 540/539, 1375/1372, 32805/32768

POTE generator: ~56/45 = 380.352
Mapping generator: ~56/45

Map: [<1 0 15 12 -7|, <0 5 -40 -29 33|]
EDOs: 41, 142, 183, 224, 631d, 855d, 1079d
Badness: 0.0315

13-limit

Commas: 540/539, 1375/1372, 4096/4095, 6656/6655

POTE generator: ~56/45 = 380.351

Map: [<1 0 15 12 -7 -15|, <0 5 -40 -29 33 59|]
EDOs: 41, 142, 183, 224, 631d, 855d
Badness: 0.0214

Octant

Commas: 32805/32768, 2259436291848/2251875390625

POTE generator: ~3/2 = 701.713
Mapping generator: ~3

Map: [<8 0 120 -117|, <0 1 -8 11|]
Wedgie: <<8 -64 88 -120 117 384||
EDOs: 24, 224, 472, 696, 1168
Badness: 0.1572

11-limit

Commas: 9801/9800, 32805/32768, 46656/46585

POTE generator: ~3/2 = 701.713
Mapping generator: ~3

Map: [<8 0 120 -117 15|, <0 1 -8 11 1|]
EDOs: 24, 224, 472, 696, 1168
Badness: 0.0448

13-limit

Commas: 729/728, 1575/1573, 2200/2197, 6656/6655

POTE generator: ~3/2 = 701.725
Mapping generator: ~3

Map: [<8 0 120 -117 15 93|, <0 1 -8 11 1 -5|]
EDOs: 24, 224, 472, 696
Badness: 0.0304